A simple derived categorical generalization of Ulrich bundles

TL;DR

This paper introduces Ulrich objects in derived categories as a generalization of Ulrich bundles, providing new characterizations and applications, including a novel approach to a longstanding question and examples beyond sheaves.

Contribution

It defines Ulrich objects in derived categories, characterizes them similarly to existing criteria, and applies this to address the Eisenbud-Schreyer question with new examples.

Findings

Ulrich objects are characterized in derived categories.

A new approach to the Eisenbud-Schreyer question is proposed.

Examples of Ulrich objects not arising from sheaves are provided.

Abstract

We define special objects, Ulrich objects, on a derived category of polarized smooth projective variety as a generalization of Ulrich bundles to the derived category. These are defined by the cohomological conditions that are the same form as a cohomological criterion determining Ulrichness for sheaves. This paper gives a characterization of the Ulrich object similar to the one in [ES03]. As an application, we have provided a new approach to the Eisenbud-Schreyer question by using the notions of the generator of the derived category. We also have given an example of Ulrich objects that are not sheaf by Yoneda extensions.